L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s − 2·9-s + 10-s + 5·11-s − 12-s + 6·13-s + 2·14-s + 15-s + 16-s − 4·17-s + 2·18-s + 19-s − 20-s + 2·21-s − 5·22-s + 23-s + 24-s − 4·25-s − 6·26-s + 5·27-s − 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s + 1.66·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.471·18-s + 0.229·19-s − 0.223·20-s + 0.436·21-s − 1.06·22-s + 0.208·23-s + 0.204·24-s − 4/5·25-s − 1.17·26-s + 0.962·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 874 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 874 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.485364934039933239775154242830, −9.021364831795416945437160207689, −8.215638924077679727031953157982, −7.03943784107861605660058665457, −6.31118726063273349133267716362, −5.72690924352723786597792403491, −4.06708156243661157750249246762, −3.30802104131748377416216167950, −1.55487791517322642430558878400, 0,
1.55487791517322642430558878400, 3.30802104131748377416216167950, 4.06708156243661157750249246762, 5.72690924352723786597792403491, 6.31118726063273349133267716362, 7.03943784107861605660058665457, 8.215638924077679727031953157982, 9.021364831795416945437160207689, 9.485364934039933239775154242830